Page:Philosophical Review Volume 20.djvu/699

685 therefore follow that they do not exist. The mathematical theory of motion employs the notions of function and variable. The effect is conceived as a function of the cause and, instead of finding a single cause for a given effect, a functional formula is determined which is made to comprehend an infinite number of causes and effects. The notion of function is bound up with mathematical deduction. In most cases of deduction the subject of the proposition is of least importance. The validity of the deduction depends uniquely on its form. Pure mathematics is not arbitrary in this assertion, for it is necessary that the hypothesis should truly imply the thesis. If we make the hypothesis that the hypothesis implies the thesis we can deduce nothing from this unless the new hypothesis truly implies the new thesis. We therefore need true propositions for the subject of implication. If we take as premises propositions which are not true the consequences would not be truly implied by the premises. This necessity for true premises involves the important distinction between a hypothesis and a premise. Rules of deduction have a double purpose, at first as premises and then as methods for deriving conclusions from hypotheses. Now if the rules of deduction were not true the conclusions derived by their means would not truly be conclusions so that we cannot derive true conclusions from false premises.

The consequences of the analysis of mathematical knowledge have an important bearing upon the theory of knowledge. Mathematics requires propositions not based on sense experience. If it is argued that mathematical truths are derived by induction, it must be remembered that sense experience can never demonstrate the principle of induction. But sense experience concerns the particular and is meaningless aside from the principle of induction. Traditional empiricism is thus refuted, but it does not therefore follow that idealism is true. Idealism—at least every theory derived from Kant—assumes that à priori truths derive their universality from the fact that they express properties of the mind. But general and à priori knowledge must possess the same objectivity that is enjoyed by the particular facts of the physical world. Logic and mathematics force us to admit a certain scholastic realism, the existence of a world of universals, which has subsistence though it does not exist in the same sense as particular objects. We have immediate knowledge of a number of propositions about universals: this is an ultimate fact. Pure mathematics, logic, is the resume of all that can be known, directly or by demonstration, about certain universals,

The present generation has come to realize that the theory of evolution does not supplant, but rather deepens, the spiritual and idealistic view of life, that we need the developed for the comprehension of the undeveloped, and that intelligence and ethical purpose are only partially explained by reference to simpler phenomena. Not only have we changed our conceptions of the